begin to destructively interfere over time. However, the fact that the frequencies have a discrete spectrum leads to another interesting result in the longer time regime; in that case, the periodic nature of the slowly varying double exponential predicts that there should also be a revival of probability at time: The revival of probability is due to the re-phasing of the various discrete frequencies. If the field were classical, the frequencies would have a continuous spectrum, and such re-phasing could never occur within a finite time. A plot of the probability to find an atom initially in the ground state to have transitioned to the excited state after interacting with a cavity prepared a in a coherent state vs. the unit-less parameter is shown to the right. Note the initial collapse followed by the clear revival at longer times. Collapses and revivals of quantum oscillations This plot of quantum oscillations of atomic inversion—for quadratic scaled detuning parameter , where is the detuning parameter—was built on the basis of formulas obtained by A.A. Karatsuba and E.A. Karatsuba. See also Caldeira–Leggett model Jaynes–Cummings–Hubbard model Rabi problem Spontaneous emission Vacuum Rabi oscillation References Further reading Quantum optics